Yes, it is natural. I had a hard time at school, and I'm a bit anxious, but purposefully trying not to let it get the better of me.
Like yours, my family moved a lot - and it was a sometimes difficult to make new friends ... again... and again ... and again. So, similar to you, I am not moving until 10 yo DS is in ... university, if then!*
Also, my mum made no effort to socialize with other parents, and in retrospect, I think that put me at a disadvantage in terms of making friends. I say this because I've noticed that it seems some parents will competitively socialize in order to ensure their children are popular. I am not willing to go that far, but I do what I can without being ... you know, one of those parents.
I volunteer as much as I can at the school - although I work, I make time as much as possible to do it. Not only because I get to see what DS is doing at school, but because I think it boosts his confidence to be able to say ... look, my mum is here.
My mum never volunteered, although she was SAHM. I have no idea what she was doing all day at home. Maybe reading a book.
Sometimes I really can't stomach the SAHMs with the bleach blond hair and perfect figures laughing and talking about their latest bathroom/kitchen renovation as if that's all they ever do ... so, I've just picked a few parents to socialize with, and that seems to help.
Unless your child is amazingly charismatic - social skills are something, I think, you kind of have to teach, and help a bit with, sometimes by role modeling it.
Does this sound like a good approach? (See, I'm not even 100% sure! Just winging it with this parenthood thing!)
*Sometimes I think it would be good for DS to have the challenge of making all new friends, but I think summer camp fills that need.
Hope that helps! Kind of long winded.